To determine the triangle where the angle [tex]\( x \)[/tex] is equal to [tex]\( \tan^{-1}\left(\frac{3.1}{5.2}\right) \)[/tex], we need to focus on the relationship between the sides of the right triangle and the angle [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Definition of the Tangent Function:
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically:
[tex]\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
2. Given Information:
We are given that:
[tex]\[
x = \tan^{-1}\left(\frac{3.1}{5.2}\right)
\][/tex]
3. Calculate [tex]\( x \)[/tex]:
From this relationship, we know that [tex]\( x \)[/tex] is the angle whose tangent is [tex]\(\frac{3.1}{5.2}\)[/tex].
4. Angle in Degrees:
The value of [tex]\( x \)[/tex] in degrees can be found using the arctangent function (inverse tangent). From the result, we know:
[tex]\[
x \approx 30.80144597613683^\circ
\][/tex]
5. Angle in Radians:
Similarly, the value of [tex]\( x \)[/tex] in radians is:
[tex]\[
x \approx 0.5375866466587464 \text{ radians}
\][/tex]
6. Identifying the Triangle:
To find the triangle with this angle, look for a right triangle where the ratio of the side lengths (opposite to [tex]\( x \)[/tex] over adjacent to [tex]\( x \)[/tex]) is [tex]\(\frac{3.1}{5.2}\)[/tex].
Thus, the triangle you are looking for is a right triangle with an angle [tex]\( x \approx 30.8^\circ \)[/tex] or [tex]\( x \approx 0.538 \text{ radians} \)[/tex] such that:
[tex]\[
\tan(x) = \frac{3.1}{5.2}
\][/tex]
You will find [tex]\( x \)[/tex] in the triangle where:
- The length of the side opposite [tex]\( x \)[/tex] is [tex]\( 3.1 \)[/tex],
- The length of the side adjacent to [tex]\( x \)[/tex] is [tex]\( 5.2 \)[/tex],
or any triangle with a proportional relationship (similar triangles) to these side lengths.
